Combining a smooth information criterion with neural networks

# Combining a smooth information criterion with neural networks
----
## **Andrew McInerney**, ** **
### University of Limerick
#### CMStatistics 2023, 17 December 2023

---

# Background

*  Research: Neural networks from a statistical-modelling perspective

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# Smooth Information Criterion

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# Smooth Information Criterion

$$
 \text{BIC} = -2\ell(\theta) + \log(n) \left[ \sum_{j=1}^p |\beta_j|^0 + 1 \right]
$$

where

`\begin{equation*}
  \ell(\theta)= -\frac{n}{2}\log(2\pi\sigma^2)-\frac{1}{2\sigma^2}\sum_{i=1}^n(y_i-x_i^T\beta)^2
\end{equation*}`

---
# Smooth Information Criterion

$$
 \text{BIC} = -2\ell(\theta) + \log(n) \left[ \sum_{j=1}^p |\beta_j|^0 + 1 \right]
$$

Introduce "smooth BIC":

$$
 \text{SBIC} = -2\ell(\theta) + \log(n) \left[ \sum_{j=1}^p \frac{{\beta_j^2}}{\beta_j^2 + \epsilon^2} + 1 \right]
$$

---
# Smooth Information Criterion

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# `$\epsilon$`-telescoping

$$
 \text{SBIC} = -2\ell(\theta) + \log(n) \left[ \sum_{j=1}^p \frac{{\beta_j^2}}{\beta_j^2 + \epsilon^2} + 1 \right]
$$

* Optimal `$\epsilon$` is zero

* Smaller `$\epsilon$` `$\implies$` less numerically stable

* Start with larger `$\epsilon$`, and "telescope" through a decreasing sequence of `$\epsilon$` values using warm starts

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# R Package

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# Extending to Neural Networks

`$$\mathbb{E}(y) = \text{NN}(X, \theta)$$`

where

`$$\text{NN}(X, \theta) = \phi_o \left[ \gamma_0+\sum_{k=1}^q \gamma_k \phi_h \left( \sum_{j=0}^p \omega_{jk}x_{j}\right) \right]$$`

---
# Extending to Neural Networks

<p style="font-size: 0.85em">
$$
 \text{SBIC} = -2\ell(\theta) + \log(n) \left[ \sum_{jk} \frac{\omega_{jk}^2}{\omega_{jk}^2 + \epsilon^2} + \sum_{k} \frac{\gamma_k^2}{\gamma_k^2 + \epsilon^2} + q + 1 \right]
$$
</p>

--
where

<p style="font-size: 1em">
$$
  \ell(\theta) = -\frac{n}{2}\log(2\pi\sigma^2)-\frac{1}{2\sigma^2}\sum_{i=1}^n(y_i-\text{NN}(x_i))^2
$$
</p>

---
# Simulation Setup

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# Results

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# Extending to Group Sparsity

.pull-left[
<img src="data:image/png;base64,#img/input-group.png" width="100%" style="display: block; margin: auto;" />
]

--
Single penalty:

`\begin{equation*}
 \frac{\omega_{jk}^2}{\omega_{jk}^2 + \epsilon^2}
\end{equation*}`

Group penalty:

$$
 \text{card}(\omega_j) \times \frac{||\omega_j||_2^2}{||\omega_j||_2^2 + \epsilon^2} 
$$

---
class: inputgroup-slide
# Group Sparsity

## Input-neuron penalization

<p style="font-size: 0.78em">
$$
 \text{IN-SBIC} = -2\ell(\theta) + \log(n) \left[ q \times \sum_{j}\frac{||\omega_j||_2^2}{||\omega_j||_2^2 + \epsilon^2} + \sum_{k} \frac{\gamma_k^2}{\gamma_k^2 + \epsilon^2} + q + 1 \right]
$$
</p>

where `$\omega_{j} = (\omega_{j1},\omega_{j2},\dotsc,\omega_{jq})^T$`

---
class: hiddengroup-slide
# Group Sparsity

## Hidden-neuron penalization

<p style="font-size: 0.78em">
$$
 \text{HN-SBIC} = -2\ell(\theta) + \log(n) \left[ (p + 1) \times \sum_{k}\frac{||\theta^{(k)}||_2^2}{||\theta^{(k)}||_2^2 + \epsilon^2} + q + 1 \right]
$$
</p>

where `$\theta^{(k)} = (\omega_{1k},\omega_{2k},\dotsc,\omega_{pk}, \gamma_k)^T$`

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# Simulation Setup

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# Results (IN-SBIC)

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# Data Application

### Insurance Data (Kaggle)

1,338 beneficiaries enrolled in an insurance plan

Response:  
`charges`

6 Explanatory Variables:

`age,` `sex,` `bmi,` `children,` `smoker,` `region`

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# Data Application - Results

---
class: selectnn-slide
# Model Selection

A Statistically-Based Approach to Feedforward Neural Network Model Selection (arXiv:2207.04248)

---
class: selectnn-slide
# Insurance: Model Selection

```r
library(selectnn)
nn <- selectnn(charges ~ ., data = insurance, Q = 8,
               n_init = 5)
summary(nn)
```

```{.bg-primary}
## [...]
## Number of input nodes: 4 
## Number of hidden nodes: 2 
## 
## Value: 1218.738 
##         Covariate Selected Delta.BIC
##        smoker.yes      Yes  2474.478
##               bmi      Yes   919.500
##               age      Yes   689.396
##          children      Yes    13.702
## [...]
```

---

Extend packages: **nnet**, **neuralnet**, **keras**, **torch**

*  Significance testing

*  Covariate-effect plots

---

```r
intnn <- interpretnn(nn)  
summary(intnn)
```

```{.bg-primary}
## Coefficients:
##                                Weights |      X^2    Pr(> X^2)
##               age     (-0.43***, 0.04) |  41.4363 1.01e-09 ***
##          sex.male        (0.08*, 0.13) |   5.5055 6.38e-02 .  
##               bmi      (0.03, 2.19***) | 105.6106 0.00e+00 ***
##          children   (-0.08***, -0.11.) |  19.0146 7.43e-05 ***
##        smoker.yes (-3.16***, -6.19***) | 250.6393 0.00e+00 ***
##  region.northwest        (0.07., 0.15) |   2.8437 2.41e-01    
##  region.southeast        (0.11*, 0.12) |   6.2560 4.38e-02 *  
##  region.southwest       (0.15**, 0.14) |  10.8218 4.47e-03 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---

```r
plotnn(intnn)
```

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# Conclusion

- Extend smooth information criterion to neural networks

- Also extend it to allow for group sparsity

- Preliminary results look good

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# Future Work

- More extensive simulation studies

- Comparison to other common approaches

- Possible extension to deep neural networks

---
class: bigger
# References

*  <font size="5">McInerney, A., & Burke, K. (2022). A statistically-based approach to feedforward neural network model selection. <i>arXiv preprint arXiv:2207.04248</i>.  </font>

*  <font size="5">McInerney, A., & Burke, K. (2023). Interpreting feedforward neural networks as statistical models. <i>arXiv preprint arXiv:2311.08139</i>.    </font>

*  <font size="5">O’Neill, M. and Burke, K. (2023). Variable selection using a smooth information criterion for distributional regression models. <i>Statistics and Computing, 33(3), p.71</i>.    </font>

### R Packages

```r
devtools::install_github(c("andrew-mcinerney/selectnn",
                           "andrew-mcinerney/interpretnn"))
```